Bonnesen-type inequalities for surfaces of constant curvature

Bonnesen-type inequalities for surfaces of constant curvature

A Bonnesen-type inequality is a sharp isoperimetric inequality that includes an error estimate in terms of inscribed and circumscribed regions. A kinematic technique is used to prove a Bonnesen-type inequality for the Euclidean sphere (having constant Gauss curvature κ > 0) and the hyperbolic plane (having constant Gauss curvature κ < 0). These generalized inequalities each converge to the clas...

Isoperimetric-type inequalities on constant curvature manifolds

By exploiting optimal transport theory on Riemannian manifolds and adapting Gromov’s proof of the isoperimetric inequality in the Euclidean space, we prove an isoperimetric-type inequality on simply connected constant curvature manifolds.

Surfaces of Constant Mean Curvature

Bonnesen-style inequality for the first eigenvalue on a complete surface of constant curvature

By Cheeger's isoperimetric constants, some lower bounds and upper bounds of [Formula: see text], the first eigenvalue on a complete surface of constant curvature, are given. Some Bonnesen-style inequalities and reverse Bonnesen-style inequalities for the first eigenvalue are obtained. Those Bonnesen-style inequalities obtained are stronger than the well-known Osserman's results and the upper bo...

New Constant Mean Curvature Surfaces

We use the DPW construction [5] to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the t...